The great switch: Relations and commutativity
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non-commutative picture

You can create:

  • A silly and profound library of non-commutative relationships
  • Your own math operations
  • Conjectures and theories about properties of your own math operations

How?
Take any action or relation between two objects. Then switch the objects. Is it all the same? The picture above has a few examples. The book reads the boy. The underwear is above the pants. Pots grow from plants. These are all examples of non-commutative relationships, when switching objects means dramatically (or comically) different results.

Most relationships in this life are non-commutative. "Sally loves Harry" can be quite different from "Harry loves Sally," as people discover early in their lives. Do you wish our world were a bit more commutative? Collect some meaningful relation and action examples, maybe in a little essay or a comic book. You can share your creations with the study group. Can you find any commutative relations at all?

If you know formal math operations, you can check them out, too. Is 2+3 the same as 3+2? How about multiplication, division, subtraction, power? Even kids who don't know formal operations yet can switch numbers in their hands-on math situations. Is it the same thing if you divide two pies among six guests, or six pies between two guests? Again, you can collect the situations as they come up in a little "math diary" or a piece of paper on your fridge.

You can take turns making up your own math operations between two numbers that do not exist yet. For example, you can add two numbers and then multiply by your second number. Give your operation a name, and a symbol, and try figuring out how to write it down in a formula. I call my operation "snail" and it works like that: 3@5=(3+5)*5 You can exchange your operation creations, and check if they are commutative.

Why?
Because this activity has to do with real life drama and can be very poetic. It is also funny, good for beginners to advanced mathematicians, and handy for car trips.

Higher and deeper

  • Commutativity plays a major role in abstract algebra and any algebra-related fields such as algebraic topology
  • There are other properties you can explore in similar manner: transitivity and distribution property come to mind. Rock, paper, scissors is a non-transitive system!
Created: December 6, 2008, 6:53 am, by MariaD
Last edit: December 6, 2008, 6:53 am, by MariaD ( Edit, History )
Co-author: MariaD